Methods for subsurface parameter estimation in full wavefield inversion and reverse-time migration

ABSTRACT

Method for converting seismic data to obtain a subsurface model of, for example, bulk modulus or density. The gradient of an objective function is computed ( 103 ) using the seismic data ( 101 ) and a background subsurface medium model ( 102 ). The source and receiver illuminations are computed in the background model ( 104 ). The seismic resolution volume is computed using the velocities of the background model ( 105 ). The gradient is converted into the difference subsurface model parameters ( 106 ) using the source and receiver illumination, seismic resolution volume, and the background subsurface model. These same factors may be used to compensate seismic data migrated by reverse time migration, which can then be related to a subsurface bulk modulus model. For iterative inversion, the difference subsurface model parameters ( 106 ) are used as preconditioned gradients ( 107 ).

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No. 61/303,148 filed Feb. 10, 2010, entitled Methods for Subsurface Parameter Estimation in Full Wavefield Inversion and Reverse-Time Migration, which is incorporated by reference, in its entirety, for all purposes.

FIELD OF THE INVENTION

This invention relates generally to the field of geophysical prospecting and, more particularly, to seismic data processing. Specifically, the invention is a method for subsurface parameter estimation in full wave field inversion and reverse-time migration.

BACKGROUND OF THE INVENTION

Full wavefield inversion (FWI) in exploration seismic processing relies on the calculation of the gradient of an objective function with respect to the subsurface model parameters [12]. An objective function E is usually given as an L₂ norm as

$\begin{matrix} {{E = {\frac{1}{2}{\int{\int{\int{{{{p\left( {r_{g},{r_{s};t}} \right)} - {p_{b}\left( {r_{g},{r_{s};t}} \right)}}}^{2}{\mathbb{d}t}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}}}}}}},} & (1) \end{matrix}$ where p and p_(b) are the measured pressure, i.e. seismic amplitude, and the modeled pressure in the background subsurface model at the receiver location r_(g) for a shot located at r_(s). In iterative inversion processes, the background medium is typically the medium resulting from the previous inversion cycle. In non-iterative inversion processes or migrations, the background medium is typically derived using conventional seismic processing techniques such as migration velocity analysis. The objective function is integrated over all time t, and the surfaces S_(g) and S_(s) that are defined by the spread of the receivers and the shots. We define K_(d)(r)=K(r)−K_(b)(r) and ρ_(d)(r)=ρ(r)−ρ_(b)(r), where K(r) and ρ(r) are the true bulk modulus and density, and K_(b)(r) and ρ_(b)(r) are the bulk modulus and the density of the background model at the subsurface location r. We also define the difference between the measured and the modeled pressure to be p_(d)(r_(g),r_(s);t)=p (r_(g),r_(s);t)−p_(b)(r_(g),r_(s);t).

The measured pressure p, satisfies the wave equation

$\begin{matrix} {\mspace{79mu}{{{{\rho\;{\nabla{\cdot \left( {\frac{1}{\rho}{\nabla p}} \right)}}} - {\frac{\rho}{K}\overset{¨}{p}}} = {{- {q(t)}}{\delta\left( {r - r_{s}} \right)}}},\mspace{79mu}{or}}} & (2) \\ {{\left( {\rho_{b} + \rho_{d}} \right)\nabla}{{{{\cdot \left( {\frac{1}{\rho_{b} + \rho_{d}}{\nabla\left( {\rho_{b} + \rho_{d}} \right)}} \right)} - {\frac{\rho_{b} + \rho_{d}}{K_{b} + K_{d}}\left( {{\overset{¨}{\rho}}_{b} + {\overset{¨}{\rho}}_{d}} \right)}} = {{- {q(t)}}{\delta\left( {r - r_{s}} \right)}}},}} & (3) \end{matrix}$ where q(t) is the source signature. By expanding the perturbation terms and keeping only the 1st order Born approximation terms, one can derive the Born scattering equation for the pressure p_(d),

$\begin{matrix} {{{{\rho_{b}{\nabla{\cdot \left( {\frac{1}{\rho_{b}}{\nabla p_{d}}} \right)}}} - {\frac{\rho_{b}}{K_{b}}{\overset{¨}{p}}_{d}}} = {- \left\lbrack {{\frac{\rho_{b}K_{d}}{K_{b}^{2}}{\overset{¨}{p}}_{b}} - {\rho_{b}{\nabla{\cdot \left( {\frac{\rho_{d}}{\rho_{b}^{2}}{\nabla p_{b}}} \right)}}}} \right\rbrack}},} & (4) \end{matrix}$ and so p_(d) satisfies

$\begin{matrix} {{{p_{d}\left( {{r_{g}r_{s}};t} \right)} = {\int{\left\lbrack {{{\rho_{b}\left( r^{\prime} \right)}\frac{K_{d}\left( r^{\prime} \right)}{K_{b}^{2}\left( r^{\prime} \right)}{{\overset{¨}{p}}_{b}\left( {r^{\prime},{r_{s};t}} \right)}} - {{\rho_{b}\left( r^{\prime} \right)}{\nabla{\cdot \left( {\frac{\rho_{d}\left( r^{\prime} \right)}{\rho_{b}^{2}\left( r^{\prime} \right)}{\nabla{p_{b}\left( {r^{\prime},{r_{s};t}} \right)}}} \right)}}}} \right\rbrack*{g_{b}\left( {r_{g},{r^{\prime};t}} \right)}{\mathbb{d}V^{\prime}}}}},} & (5) \end{matrix}$ where V′ is the volume spanned by r′, and g_(b) is the Green's function in the background medium.

One can derive the equations for the gradients of p_(b) using Eq. (5) and by considering the fractional change δp_(b) due to fractional change δK_(b) and δρ_(b) over an infinitesimal volume dV,

$\begin{matrix} {{\frac{\partial p_{b}}{\partial{K_{b}(r)}} = {\frac{{\rho_{b}(r)}{dV}}{K_{b}^{2}(r)}{g_{b}\left( {r_{g},{r;t}} \right)}*{{\overset{¨}{p}}_{b}\left( {r,{r_{s};t}} \right)}}},{and}} & (6) \\ {{\frac{\partial p_{b}}{\partial{\rho_{b}(r)}} = {F^{- 1}\left\{ {\frac{dV}{\rho_{b}(r)}{{\nabla{G_{b}\left( {r_{g},{r;f}} \right)}} \cdot {\nabla{P_{b}\left( {r,{r_{s};f}} \right)}}}} \right\}}},} & (7) \end{matrix}$ where P_(b)=F{x}, P_(d)=F{p_(d)}, G_(b)=F{g_(b)}, and F and F⁻¹ are the Fourier transform and the inverse Fourier transform operators.

By using Eqs. 6 and 7, and using the reciprocity relationship ρ_(b)(r)G_(b)(r_(g),r)=ρ_(b)(r_(g))G_(b)(r,r_(g)),

$\begin{matrix} {\begin{matrix} {\frac{\partial E}{\partial{K_{b}(r)}} = {- {\int{\int{\int{p_{d}\frac{\partial p_{b}}{\partial{K_{b}(r)}}{\mathbb{d}t}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}}}}}}} \\ {= {{- \frac{{\rho_{b}(r)}{dV}}{K_{b}^{2}(r)}}{\int{\int{\int{\left( {i\; 2\;\pi\; f} \right)^{2}{P_{b}\left( {r,{r_{s};f}} \right)}}}}}}} \\ {{G_{b}\left( {r_{g},{r;f}} \right)}{P_{d}^{*}\left( {r_{g},{r_{s};f}} \right)}{\mathbb{d}f}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}} \\ {= {\frac{{\rho_{b}(r)}{dV}}{K_{b}^{2}(r)}{\int{\int{{{\overset{.}{p}}_{b}\left( {r,{r_{s};t}} \right)}{\int{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{g_{b}\left( {r,{r_{g};{- t}}} \right)}*}}}}}}} \\ {{{{\overset{.}{p}}_{d}\left( {r_{g},{r_{s};t}} \right)}{\mathbb{d}S_{g}}{\mathbb{d}t}{\mathbb{d}S_{s}}},} \end{matrix}\mspace{79mu}{and}} & (8) \\ \begin{matrix} {\frac{\partial E}{\partial{\rho_{b}(r)}} = {\int{\int{\int{p_{d}\frac{\partial p_{b}}{\partial{\rho_{b}(r)}}{\mathbb{d}t}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}}}}}} \\ {= {{- \frac{dV}{\rho_{b}(r)}}{\int{\int{\int{{P_{d}^{*}\left( {r_{g},{r_{s};f}} \right)}{{\nabla{P_{b}\left( {r,{r_{s};f}} \right)}} \cdot}}}}}}} \\ {{\nabla{G_{b}\left( {{r_{g}r};f} \right)}}{\mathbb{d}f}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}} \\ {= {{- \frac{dV}{\rho_{b}(r)}}{\int{\int{{\nabla{p_{b}\left( {r,{r_{s};t}} \right)}} \cdot}}}}} \\ {\left\lbrack {\int{{\nabla\left( {{g_{b}\left( {r,{r_{g};{- t}}} \right)}*\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{p_{d}\left( {r_{g},{r_{s};t}} \right)}} \right)}{\mathbb{d}S_{g}}}} \right\rbrack{\mathbb{d}t}{{\mathbb{d}S_{s}}.}} \end{matrix} & (9) \end{matrix}$ One can then use Eqs. 8 and 9 to perform full wavefield inversion in an iterative manner.

Reverse-time migration (RTM) is based on techniques similar to gradient computation in FWI, where the forward propagated field is cross-correlated with the time-reversed received field. By doing so, RTM overcomes limitations of ray-based migration techniques such as Kirchhoff migration. In RTM, the migrated image field M at subsurface location r is given as M(r)=∫∫p _(b)(r,r _(s) ;t)∫g _(b)(r,r _(g) ;−t)*p(r _(g) ,r _(s) ;t)dS _(g) dtdS _(s),  (10) which is very similar to the gradient equation 8 of FWI.

While Eqs. 8 and 9 provide the framework for inverting data into subsurface models, the convergence of the inversion process often is very slow. Also, RTM using Eq. 10 suffers weak amplitude in the deep section due to spreading of the wavefield. Many attempts have been made to improve the convergence of FWI or improve the amplitude of the reverse-time migration by using the Hessian of the objective function [9], i.e., a second derivative of the objective function. Computation of the Hessian, however, is not only prohibitively expensive in computational resources, but it requires prohibitively large storage space for a realistic 3-D inversion problem. Furthermore, FWI using the full Hessian matrix may result in suboptimal inversion [2].

One may be able to perform more stable inversion by lumping non-diagonal terms of the Hessian into the diagonal terms [2]. These, however, still require computation of the full Hessian matrix or at least a few off-diagonal terms of the Hessian matrix, which can be costly computations. While one may choose to use the diagonal of the Hessian only [11], this is valid only in the high frequency asymptotic regime with infinite aperture [1, 7].

Plessix and Mulder tried to overcome these difficulties by first computing an approximate diagonal Hessian, then by scaling these by z^(α)ν_(p) ^(β), where z is the depth and ν_(p) is the compressional wave velocity [7]. From numerical experiment, they have determined that the best scaling parameter is z^(0.5)ν_(p) ^(0.5). This approach, however, does not provide quantitative inversion of the subsurface medium parameters with correct units, since only approximate scaling has been applied. Furthermore, this approach was applied to RTM where only variations in compressional wave velocity is considered, and so may not be applicable to FWI where other elastic parameters such as density and shear wave velocity vary in space.

SUMMARY OF THE INVENTION

In one embodiment, the invention is a method for determining a model of a physical property in a subsurface region from inversion of seismic data, acquired from a seismic survey of the subsurface region, or from reverse time migration of seismic images from the seismic data, said method comprising determining a seismic resolution volume for the physical property and using it as a multiplicative scale factor in computations performed on a computer to either

(a) convert a gradient of data misfit in an inversion, or

(b) compensate reverse-time migrated seismic images,

to obtain the model of the physical property or an update to an assumed model.

In some embodiments of the inventive method, the gradient of data misfit or the reverse time migrated seismic images are multiplied by additional scale factors besides seismic resolution volume, wherein the additional scale factors include a source illumination factor, a receiver illumination factor, and a background medium properties factor. This results in a model of the physical property or an update to an assumed model having correct units.

It will be obvious to those who work in the technical field that in any practical application of the invention, inversion or migration of seismic data must be performed on a computer specifically programmed to carry out that operation.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention and its advantages will be better understood by referring to the following detailed description and the attached drawings in which:

FIG. 1 is a flowchart showing basic steps in one embodiment of the present inventive method;

FIGS. 2-5 pertain to a first example application of the present invention, where FIG. 2 shows the gradient of the objective function with respect to the bulk modulus in Pa m⁴ s, computed using Eq. 8;

FIG. 3 shows the bulk modulus update

K_(d) (r)

in Pa computed using Eq. 18 and the gradient in FIG. 2;

FIG. 4 shows the bulk modulus update

K_(d)(r)

in Pa computed using Eq. 24 and the gradient in FIG. 2;

FIG. 5 shows the gradient of the objective function with respect to the density in Pa² m⁷ s/kg, computed using Eq. 9;

FIGS. 6 and 7 pertain to a second example application of the present invention, where FIG. 6 shows the density update

ρ_(d)(r)

in kg/m³ computed using Eq. 28 and the gradient in FIG. 5; and

FIG. 7 shows the density update

ρ_(d)(r)

in kg/m³ computed using Eq. 34 and the gradient in FIG. 5.

The invention will be described in connection with example embodiments. However, to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention, this is intended to be illustrative only, and is not to be construed as limiting the scope of the invention. On the contrary, it is intended to cover all alternatives, modifications and equivalents that may be included within the scope of the invention, as defined by the appended claims.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

We derive inversion equations for K_(d) and ρ_(d) in the present invention using Eqs. 8 and 9. This is done by first taking advantage of the fact that p_(d) in Eqs. 8 and 9 can also be expanded using the Born approximation in Eq. 5. Neglecting the crosstalk components between K_(d) and ρ_(d), Eqs. 8 and 9 can then be approximated as

$\begin{matrix} {\frac{\partial E}{\partial{K_{b}(r)}} \approx {{- \frac{{\rho_{b}(r)}{dV}}{K_{b}^{2}(r)}}{\int{\int{\int{\int{{\frac{{\rho_{b}\left( r^{\prime} \right)}{K_{d}\left( r^{\prime} \right)}}{K_{b}^{2}\left( r^{\prime} \right)}\left\lbrack {{g_{b}\left( {r_{g},{r^{\prime};t}} \right)}*{{\overset{¨}{p}}_{b}\left( {r^{\prime},{r_{s};t}} \right)}} \right\rbrack} \times {\quad{{\left\lbrack {{g_{b}\left( {r_{g},{r;t}} \right)}*{{\overset{¨}{p}}_{b}\left( {r,{r_{s};t}} \right)}} \right\rbrack{\mathbb{d}t}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}{\mathbb{d}V^{\prime}}},\mspace{79mu}{and}}}}}}}}}} & (11) \\ {\frac{\partial E}{\partial{\rho_{b}(r)}} \approx {\frac{dV}{\rho_{b}(r)}{\int{\int{\int{\int{{\frac{\rho_{d}\left( r^{\prime} \right)}{\rho_{b}\left( r^{\prime} \right)}\left\lbrack {{\nabla^{\prime}{P_{b}^{*}\left( {r^{\prime},{r_{s};f}} \right)}} \cdot {\nabla^{\prime}{G_{b}^{*}\left( {r_{g},{r^{\prime};f}} \right)}}} \right\rbrack} \times {\quad{\left\lbrack {{\nabla{P_{b}\left( {r,{r_{s};f}} \right)}} \cdot {\nabla{G_{b}\left( {r_{g},{r;f}} \right)}}} \right\rbrack{\mathbb{d}t}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}{{\mathbb{d}V^{\prime}}.}}}}}}}}}} & (12) \end{matrix}$ By changing the orders of the integral, Eq. 11 can be rewritten in the frequency domain as

$\begin{matrix} {\frac{\partial E}{\partial{K_{b}(r)}} \approx {\frac{{\rho_{b}(r)}{dV}}{K_{b}^{2}(r)}{\int{\int{\left( {2\;\pi\; f} \right)^{4}\frac{{\rho_{b}\left( r^{\prime} \right)}{dV}\mspace{14mu}{K_{d}\left( r^{\prime} \right)}}{K_{b}^{2}\left( r^{\prime} \right)} \times {\quad{\left\{ {\int{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{G_{b}\left( {r,{r_{g};f}} \right)}{G_{b}^{*}\left( {r_{g},{r^{\prime};f}} \right)}{\mathbb{d}S_{g}}}} \right\}\left\{ {\int{{P_{b}\left( {r,{r_{s};f}} \right)}{P_{b}^{*}\left( {r^{\prime},{r_{s};f}} \right)}{\mathbb{d}S_{s}}}} \right\}{\mathbb{d}V^{\prime}}{{\mathbb{d}f}.}}}}}}}} & (13) \end{matrix}$ The first integral term

$\begin{matrix} {\int{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{G_{b}\left( {r,{r_{g};f}} \right)}{G_{b}^{*}\left( {r_{g},{r^{\prime};f}} \right)}{\mathbb{d}S_{g}}}} & (14) \end{matrix}$ in Eq. 13 is the approximation to the time reversal backpropagation for a field generated by an impulse source at r′, measured over the surface S_(g), and then backpropagated to r (See, e.g., Refs. [8, 3]). The wavefield due to this term propagates back towards the impulse source location at r′, and behaves similar to the spatial delta function δ(r−r′) when t=0, if the integral surface S_(g) embraces the point r′. This wavefield is correlated with the wavefield due to the second term, ∫P_(b)(r,r_(s); f)P_(b)*(r′,r_(s); f)dS_(s), to form the gradient near r=r′. The correlation of the first and the second term then decays rapidly near r=r′. The present invention recognizes that the zone in which the amplitude of the correlation term is not negligible is determined by the seismic resolution of the survey. In the present invention, we make an approximation that

$\begin{matrix} {{{\int{\left( {2\pi\; f} \right)^{4}\left\{ {\int{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{G_{b}\left( {r,{r_{g};f}} \right)}{G_{b}^{*}\left( {r_{g},{r^{\prime};f}} \right)}{\mathbb{d}S_{g}}}} \right\}\left\{ {\int{{P_{b}\left( {r,{r_{s};f}} \right)}{P_{b}^{*}\left( {r^{\prime},{r_{s};f}} \right)}{\mathbb{d}S_{s}}}} \right\}{\mathbb{d}f}}} \approx {{I_{K}(r)}{V_{K}(r)}{\delta\left( {r - r^{\prime}} \right)}}},} & (15) \end{matrix}$ where

$\begin{matrix} {{{I_{K}(r)} = {\int{\left( {2\pi\; f} \right)^{4}\left\{ {\int{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{G_{b}\left( {r,{r_{g};f}} \right)}{G_{b}^{*}\left( {r_{g},{r;f}} \right)}{\mathbb{d}S_{g}}}} \right\} \times \left\{ {\int{{P_{b}\left( {r,{r_{s};f}} \right)}{P_{b}^{*}\left( {r,{r_{s};f}} \right)}{\mathbb{d}S_{s}}}} \right\}{\mathbb{d}f}}}},} & (16) \end{matrix}$ and V_(K) (r) is the seismic resolution at the subsurface location r. Equation 15 is equivalent to the mass lumping of the Gauss-Newton Hessian matrix by assuming that the non-diagonal components are equal to the diagonal components when the non-diagonal components are within the seismic resolution volume of the diagonal component, and those outside the resolution volume zero. In other words, Eq. 15 is equivalent to implicitly counting the number of non-diagonal components N_(i) of the Gauss-Newton Hessian matrix in each i-th row that are significant in amplitude by using the seismic resolution volume of the survey, then multiplying the diagonal component of the i-th row by N_(i).

Seismic resolution volume may be thought of as the minimal volume at r that a seismic imaging system can resolve under given seismic data acquisition parameters. Two small targets that are within one seismic resolution volume of each other are usually not resolved and appear as one target in the seismic imaging system. The resolution volumes for different medium parameters are different due to the difference in the radiation pattern. For example, the targets due to a bulk modulus perturbation yield a monopole radiation pattern, while those due to a density perturbation yield a dipole radiation pattern. Seismic resolution volume V_(K)(r) for bulk modulus can be computed, for example, using a relatively inexpensive ray approximation [6, 4]. Persons who work in the technical field may know other ways to estimate the resolution volume. For example, one may be able to empirically estimate the resolution volume by distributing point targets in the background medium, and by investigating spread of the targets in the seismic image. If the background medium contains discontinuity in velocity due to iterative nature of the inversion, the background medium may need to be smoothed for ray tracing. One may also make a simplifying assumption that the wavenumber coverage is uniform. The seismic resolution volume in this case is a sphere with radius σ≈(5/18π)^(0.5)ν_(p)(r)/f_(p), where f_(p) is the peak frequency [6]. One may also employ an approximation, σ≈ν_(p)(r)T/4=ν_(p)(r)/4B, where T and B are the effective time duration and the effective bandwidth of the source waveform, following the radar resolution equation [5].

Equation 11 can then be simplified using Eq. 15 as

$\begin{matrix} {{\frac{\partial E}{\partial{K_{b}(r)}} \approx {{- \frac{{\rho_{b}^{2}(r)}V}{K_{b}^{4}(r)}}\left\langle {K_{d}(r)} \right\rangle{I_{K}(r)}{V_{K}(r)}}},} & (17) \\ {{and}\mspace{14mu}{so}} & \; \\ {{\left\langle {K_{d}(r)} \right\rangle \approx {{- \frac{K_{b}^{4}(r)}{{\rho_{b}^{2}(r)}{dV}}}\frac{1}{{I_{K}(r)}{V_{K}(r)}}\frac{\partial E}{\partial{K_{b}(r)}}}},} & (18) \end{matrix}$ where

K_(d) (r)

is the spatial average of K_(d) over the seismic resolution at spatial location r.

Equation 16 can be further simplified if we use free-space Green's function

$\begin{matrix} {{{G\left( {r_{g},{r;f}} \right)} = {\frac{1}{4\pi{{r - r_{g}}}}e^{{\mathbb{i}}\; k{{r - r_{g}}}}}},} & (19) \end{matrix}$ and assume that S_(g) subtends over half the solid angle. Equation 16 then simplifies to

$\begin{matrix} {{{I_{K}(r)} \approx {{I_{K,s}(r)}{I_{K,g}(r)}}},{where}} & (20) \\ {{{I_{K,s}(r)} = {\int{{{{\overset{¨}{p}}_{b}\left( {r,{r_{s};t}} \right)}}^{2}{\mathbb{d}t}}}},{and}} & (21) \\ {{I_{K,g}(r)} = {{\int{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{G_{b}\left( {r,{r_{g};f}} \right)}{G_{b}^{*}\left( {r_{g},{r;f}} \right)}{\mathbb{d}S_{g}}}} \approx {\frac{1}{8\pi}{\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}.}}}} & (22) \end{matrix}$ The term I_(K,s)(r) may be recognized as the source illumination in the background model, and I_(K,g)(r) can be understood to be the receiver illumination. One may also be able to vary the solid angle of integral at each subsurface location r following the survey geometry. Equation 11 then becomes

$\begin{matrix} {\frac{\partial E}{\partial{K_{b}(r)}} \approx {{- \frac{{\rho_{b}^{2}(r)}{dV}}{K_{b}^{4}(r)}}\left\langle {K_{d}(r)} \right\rangle{I_{K,s}(r)}{I_{K,g}(r)}{V_{K}(r)}}} & (23) \\ {and} & \; \\ {\left\langle {K_{d}(r)} \right\rangle \approx {{- \frac{K_{b}^{4}(r)}{{\rho_{b}^{2}(r)}{dV}}}\frac{1}{{I_{K,s}(r)}{I_{K,g}(r)}{V_{K}(r)}}{\frac{\partial E}{\partial{K_{b}(r)}}.}}} & (24) \end{matrix}$

Equations 18 and 24 show that one can convert the gradient ∂E/∂K_(b)(r) into a medium parameter

K_(d)(r)

by scaling the gradient by the source and receiver illumination, resolution volume, and the background medium properties. If the inversion process is not iterative, one should be able to use Eq. 24 for parameter inversion. If the inversion process is iterative, one can use

K_(d)(r)

in Eq. 24 as a preconditioned gradient for optimization techniques such as steepest descent, conjugate gradient (CG), or Newton CG method. It is important to note that Eqs. (18) and (24) yield bulk modulus with the correct units, i.e. are dimensionally correct, because all terms have been taken into account, and none have been neglected to simplify the computation as is the case with some published approaches. The published approaches that neglect one or more of the terms source illumination, receiver illumination, background medium properties and seismic resolution volume, will not result in the correct units, and therefore will need some sort of ad hoc fix up before they can be used for iterative or non-iterative inversion.

For density gradient, we make an assumption similar to that in Eq. 15, ∫∫[∇′P _(b)*(r′,r _(s) ;f)·∇′G _(b)*(r _(g) ,r′;f)]×[∇P _(b)(r,r _(s) ;f)·∇G _(b)(r _(g) ,r;f)]dS _(g) dS _(s) df≈I _(ρ)(r)V _(ρ)(r)δ(r−r′),  (25) where I _(ρ)(r)=∫∫∫|∇P _(b)(r,r _(s) ;f)·∇G _(b)(r _(g) ,r;f)|² dS _(g) dS _(s) df,  (26) and V_(ρ)(r) is the seismic resolution for density ρ at subsurface location r. The resolution volume V_(ρ)(r) differs from V_(K)(r), since the wavenumbers are missing when the incident and the scattered field are nearly orthogonal to each other. This is due to the dipole radiation pattern of the density perturbation, as discussed previously. As was done for V_(K)(r), one can employ ray tracing to compute the resolution volume V_(ρ)(r) while accounting for these missing near-orthogonal wavenumbers. Alternatively, one may be able assume V_(K)(r)≈V_(ρ)(r) by neglecting the difference in the wavenumber coverage.

The gradient equation 12 can then be rewritten as

$\begin{matrix} {{\frac{\partial E}{\partial{\rho_{b}(r)}} \approx {{- \frac{dV}{\rho_{b}^{2}(r)}}\left\langle {\rho_{d}(r)} \right\rangle{I_{\rho}(r)}{V_{\rho}(r)}}},} & (27) \\ {{and}\mspace{14mu}{so}} & \; \\ {{\left\langle {\rho_{d}(r)} \right\rangle \approx {{- \frac{\rho_{b}^{2}(r)}{dV}}\frac{1}{{I_{\rho}(r)}{V_{\rho}(r)}}\frac{\partial E}{\partial{\rho_{b}(r)}}}},} & (28) \end{matrix}$ where

ρ_(d)(r)

is the spatial average of ρ_(d)(r) over the seismic resolution volume V_(ρ)(r).

We can further simplify Eq. 25 by using the vector identity (a·b)(c·d)=(a·d)(b·c)+(a×c)·(b×d) to obtain [∇′P _(b)*(r′,r _(s) ;f)·∇′G _(b)*(r _(g) ,r′;f)][∇P _(b)(r,r _(s) ;f)·∇G _(b)(r _(g) ,r;f)]=[∇′P _(b)*(r′,r _(s) ;f)·∇P _(b)(r,r _(s) ;f)][∇′G _(b)*(r _(g) ,r′;f)·∇G _(b)(r _(g) ,r;f)]+[∇′P _(b)*(r′,r _(s) ;f)×∇G _(b)(r _(g) ,r;f)]·[∇′G _(b)*(r _(g) ,r′;f)×∇P _(b)(r,r _(s) ;f)].  (29) The second term in the right-hand side of Eq. 29 is the correction term for the dipole radiation pattern of the scattered field, and so it reaches a maximum when ∇P_(b) and ∇G_(b) are orthogonal to each other. Neglecting this correction term, [∇′P _(b)*(r′,r _(s) ;f)·∇′G _(b)*(r _(g) ,r′;f)][∇P _(b)(r,r _(s) ;f)·∇G _(b)(r _(g) ,r;f)]≈[∇′P _(b)*(r′,r _(s) ;f)·∇P _(b)(r,r _(s) ;f)][∇′G_(b)*(r _(g) ,r′;f)·∇G _(b)(r _(g) ,r;f)].  (30) Then Eq. 12 can be approximated as

$\begin{matrix} {\frac{\partial E}{\partial{\rho_{b}(r)}} \approx {{- \frac{dV}{\rho_{b}(r)}}{\int{\int{\frac{\rho_{b}\left( r^{\prime} \right)}{\rho_{b}\left( r^{\prime} \right)}\left\{ {\int{\left\lbrack {{\nabla^{\prime}{G_{b}^{*}\left( {r_{g},{r^{\prime};f}} \right)}} \cdot {\nabla{G_{b}\left( {r_{g},{r;f}} \right)}}} \right\rbrack{\mathbb{d}S_{g}}}} \right\} \times \left\{ {\int{\left\lbrack {{\nabla^{\prime}{P_{b}^{*}\left( {r^{\prime},{r_{s};f}} \right)}} \cdot {\nabla{P_{b}\left( {r,{r_{s};f}} \right)}}} \right\rbrack{\mathbb{d}S_{s}}}} \right\}{\mathbb{d}f}{\mathbb{d}V^{\prime}}}}}}} & (31) \end{matrix}$ We can approximate the integral over S_(g) in Eq. 26 by using free-space Green's function,

$\begin{matrix} {{{\int{{{\nabla{G_{b}^{*}\left( {r_{g},{r;f}} \right)}} \cdot {\nabla{G_{b}\left( {r_{g},{r;f}} \right)}}}{\mathbb{d}S_{g}}}} \approx {\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{\int{{{\nabla{G_{b}^{*}\left( {r_{g},{r;f}} \right)}} \cdot {\nabla{G_{b}\left( {r,{r_{g};f}} \right)}}}{\mathbb{d}S_{g}}}}} \approx {\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}\frac{\left( {2\pi\; f} \right)^{2}}{8\pi\;{v_{p}^{2}(r)}}}},} & (32) \end{matrix}$ The integral was performed over half the solid angle under the assumption that ∇(ρ_(b)(r_(g))/ρ_(b)(r))≈0, and ρ_(b) (r_(g)) is constant along S_(g).

The gradient equation above can then be rewritten as

$\begin{matrix} {\frac{\partial E}{\partial{\rho_{b}(r)}} \approx {{- \frac{dV}{\rho_{b}^{2}(r)}}\left\langle {\rho_{d}(r)} \right\rangle{I_{\rho,s}(r)}{I_{\rho,g}(r)}{V_{\rho}(r)}}} & (33) \\ {{and}\mspace{14mu}{so}} & \; \\ {{\left\langle {\rho_{d}(r)} \right\rangle \approx {{- \frac{\rho_{b}^{2}(r)}{dV}}\frac{1}{{I_{\rho,s}(r)}{I_{\rho,g}(r)}{V_{\rho}(r)}}\frac{\partial E}{\partial{\rho_{b}(r)}}}},} & (34) \\ {where} & \; \\ {{{I_{\rho,s}(r)} = {\int{{{\nabla{\overset{.}{p}\left( {r,{r_{s};t}} \right)}}}^{2}{\mathbb{d}t}{\mathbb{d}S_{s}}}}},} & (35) \\ {and} & \; \\ {{I_{\rho,s}(r)} = {\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{\frac{1}{8\pi\;{v_{p}^{2}(r)}}.}}} & (36) \end{matrix}$ As was the case with

K_(d)(r)

, Eqs. 28 or 34 can be used as an inversion formula for non-iterative inversion, or as a preconditioned gradient equation for iterative inversion. It is important to note that these equations yield density with the correct units, i.e. are dimensionally correct, because all terms have been taken into account, and none have been neglected to simplify the computation as is the case with some published approaches. The same is true for Eqs. 18 and 24 for bulk modulus. The published approaches that neglect one or more of the terms source illumination, receiver illumination, background medium properties and seismic resolution volume, will not result in the correct units, and therefore will need some sort of ad hoc fix up before they can be used for iterative or non-iterative inversion.

Since the first iteration of FWI is similar to RTM, the method provided here can be applied to analyze the amplitude term in RTM with little modification. Seismic migration including RTM is typically used to image the structure of the subsurface, and so amplitude information in the migrated image is often discarded. We show that the RTM amplitude, when properly scaled using the method provided here, represents the difference between the true compressional wave velocity of the subsurface and the velocity of the background model.

We note that the RTM equation 10 is missing the double derivative of the incident field in Eq. 8. This double derivative represents that high frequency components scatter more efficiently than the low frequency components in the classical Rayleigh scattering regime (Refs. [10, 13]). Therefore, we can consider Eq. 10 as the gradient computation operation in Eq. 8, partially neglecting the frequency dependence of the scattered field,

$\begin{matrix} {{\frac{\partial E}{\partial{K_{b}(r)}} \approx {{- \frac{\rho_{b}{{dV}\left( {{\mathbb{i}}\; 2\pi\; f_{c}} \right)}^{2}}{K_{b}^{2}(r)}}{\int{\int{\int{{P_{b}\left( {r,{r_{s};f}} \right)}{G_{b}\left( {r,{r_{g};f}} \right)}{P^{*}\left( {r_{g},{r_{s};f}} \right)}{\mathbb{d}f}{\mathbb{d}S_{g}}{\mathbb{d}S_{s}}}}}}} \approx {{- \frac{\rho_{b}{{dV}\left( {{\mathbb{i}}\; 2\pi\; f_{c}} \right)}^{2}}{K_{b}^{2}(r)}}{M(r)}}},} & (37) \end{matrix}$ where f_(c) is the center frequency of the source waveform. Frequency dependence is partially neglected because, while the frequency dependence in the forward field p_(b) has been neglected, the frequency dependence implicit in the received field p_(s) cannot be neglected. Spatial variation of density is usually not considered in RTM, and so ρ_(b) is assumed to be constant in Eq. 37.

One can now apply the same approximation used to derive Eqs. 17 and 23 to Eq. 10,

$\begin{matrix} {{\frac{\partial E}{\partial{K_{b}(r)}} \approx {{- \frac{\rho_{b}^{2}{I_{K,g}(r)}{V_{K}(r)}{dV}\left\langle {K_{d}(r)} \right\rangle\left( {{\mathbb{i}2\pi}\; f_{c}} \right)^{2}}{K_{b}^{4}(r)}}{\int{\int{\left( {{\mathbb{i}}\; 2\pi\; f} \right)^{2}{P_{b}\left( {r,{r_{s};f}} \right)}{P_{b}^{*}\left( {r,{r_{s};f}} \right)}{\mathbb{d}f}{\mathbb{d}S_{s}}}}}}},} & (38) \end{matrix}$ which, together with Eq. 37, yields

$\begin{matrix} {\left\langle {K_{d}(r)} \right\rangle \approx {\frac{K_{b}^{4}(r)}{\rho_{b}{I_{K,g}(r)}{V_{K}(r)}}{\frac{M(r)}{\int{\int{{{{\overset{.}{p}}_{b}\left( {r,{r_{s};t}} \right)}}^{2}{\mathbb{d}t}{\mathbb{d}S_{s}}}}}.}}} & (39) \end{matrix}$ Equation 39 enables quantitative analysis of the amplitude in the reverse-time migrated image. More specifically, it enables inversion of the amplitude into the difference bulk modulus of the subsurface.

FIG. 1 is a flowchart showing basic steps in one embodiment of the present inventive method. In step 103, the gradient of an objective function is computed using an input seismic record (101) and information about the background subsurface medium (102). At step 104, the source and receiver illumination in the background model is computed. At step 105, the seismic resolution volume is computed using the velocities of the background model. At step 106, the gradient from step 103 is converted into the difference subsurface model parameters using the source and receiver illuminations from step 104, seismic resolution volume from step 105, and the background subsurface model (102). If an iterative inversion process is to be performed, at step 107 the difference subsurface model parameters from step 106 are used as preconditioned gradients for the iterative inversion process.

EXAMPLES

We consider the case of a 30 m×30 m×30 m “perfect” Born scatterer in a homogeneous medium with K_(b)=9 MPa and ρ_(b)=1000 kg/m³. The target is centered at (x,y,z)=(0,0,250 m), where x and y are the two horizontal coordinates and z is the depth. The target may be seen in FIGS. 2-7 as a 3×3 array of small squares located in the center of each diagram. We assume co-located sources and receivers in the −500 m≦x≦500 m and −500 m≦y≦500 m interval with the 10-m spacing in both x and y directions. We assume that the source wavelet has a uniform amplitude of 1 Pa/Hz at 1 m in the 1 to 51 Hz frequency band.

In the first example, we assume that the target has the bulk modulus perturbation given as K_(d)=900 kPa. FIG. 2 shows the gradient ∂E/∂K_(b)(r) along the y=0 plane using Eq. 8. The scattered field p_(d) in Eq. 8 has been computed using Eq. 5. The gradient in FIG. 2 has units of Pa m⁴ s, and so cannot be directly related to K_(d). As described in the “Background” section above, this is a difficulty encountered in some of the published attempts to compute a model update from the gradient of the objective function.

FIG. 3 shows

K_(d)(r)

using Eq. 18 of the present invention. It has been assumed that the seismic resolution volume V_(K)(r) is a sphere with radius σ=ν_(p)(r)/4B=15 m. One can see that FIG. 3 is the smeared image of the target, since

K_(d)(r)

is the averaged property over the seismic resolution volume. FIG. 4 is

K_(d) (r)

when the less rigorous Eq. 24 is employed. One can see that

K_(d)(r)

in FIGS. 3 and 4 are in good agreement with each other. The value of

K_(d)(r)

at the center of the target in FIG. 3 is 752 kPa, and that in FIG. 4 is 735 kPa, both of which are within 20% of the true value of 900 kPa.

The second example is the case where the target has a density perturbation of ρ_(d)=100 kg/m³. FIG. 5 shows the gradient ∂E/∂ρ_(b)(r) along the y=0 plane using Eq. 9. The scattered field in Eq. 9 has been computed using Eq. 5. The gradient in FIG. 5 has units of Pa² m⁷ s/kg. As with the first example, the different units prevent the gradient from being directly related to a density update, again illustrating the problem encountered in published methods.

FIG. 6 shows

ρ_(d)(r)

using Eq. 28. The seismic resolution volume V_(ρ)(r) has been assumed to be identical to V_(K)(r). FIG. 7 is

ρ_(d)(r)

when Eq. 34 is employed. Estimation of

ρ_(d)(r)

using Eq. 34 results in less accurate inversion than that using Eq. 28 due to the neglected dipole illumination term in Eq. 31.

The foregoing patent application is directed to particular embodiments of the present invention for the purpose of illustrating it. It will be apparent, however, to one skilled in the art, that many modifications and variations to the embodiments described herein are possible. All such modifications and variations are intended to be within the scope of the present invention, as defined in the appended claims. Persons skilled in the art will readily recognize that in practical applications of the invention, at least some of the steps in the present inventive method are performed on or with the aid of a computer, i.e. the invention is computer implemented.

REFERENCES

-   [1] G. Beylkin, “Imaging of discontinuities in the inverse     scattering problem by inversion of a causal generalized Radon     transform,” J. Math. Phys. 26, 99-108 (1985). -   [2] G. Chavent and R.-E. Plessix, “An optimal true-amplitude     least-squares prestack depth-migration operator,” Geophysics 64(2),     508-515 (1999). -   [3] D. R. Jackson and D. R. Dowling, “Phase conjugation in     underwater acoustics,” J. Acoust. Soc. Am. 89(1), 171-181 (1991). -   [4] I. Lecomte, “Resolution and illumination analyses in PSDM: A     ray-based approach,” The Leading Edge, pages 650-663 (May 2008). -   [5] N. Levanon, Radar Principles, chapter 1, pages 1-18, John Wiley     & Sons, New York (1988). -   [6] M. A. Meier and P. J. Lee, “Converted wave resolution,”     Geophysics 74(2), Q1-Q16 (2009). -   [7] R. E. Plessix and W. A. Mulder, “Frequency-domain     finite-difference amplitude-preserving migration,” Geophys. J. Int.     157, 975-987 (2004). -   [8] R. P. Porter, “Generalized holography with application to     inverse scattering and inverse source problems,” Progress in Optics     XXVII, E. Wolf, editor, pages 317-397, Elsevier (1989). -   [9] R. G. Pratt, C. Shin, and G. J. Hicks, “Gauss-Newton and full     Newton methods in frequency-space seismic waveform inversion,”     Geophys. J. Int. 133, 341-362 (1998). -   [10] J. W. S. Rayleigh, “On the transmission of light through an     atmosphere containing small particles in suspension, and on the     origin of the blue of the sky,” Phil. Mag. 47, 375-384 (1899). -   [11] C. Shin, S. Jang, and D.-J. Min, “Waveform inversion using a     logarithmic wavefield,” Geophysics 49, 592-606 (2001). -   [12] A. Tarantola, “Inversion of seismic reflection data in the     acoustic approximation,” Geophysics 49, 1259-1266 (1984). -   [13] R. J. Urick, Principles of Underwater Sound, chapter 9, pages     291-327, McGraw-Hill, New York, 3rd edition (1983). 

The invention claimed is:
 1. A method for determining a model of a physical property in a subsurface region from inversion of seismic data, acquired from a seismic survey of the subsurface region, or from reverse time migration of seismic images from the seismic data, said method comprising determining a seismic resolution volume for the physical property and using it as a multiplicative scale factor in computations performed on a computer to either convert a gradient of data misfit in an inversion, or compensate reverse-time migrated seismic images, to obtain the model of the physical property or an update to an assumed model.
 2. The method of claim 1, further comprising multiplying the gradient of data misfit or the reverse time migrated seismic images by additional scale factors including a source illumination factor, a receiver illumination factor, and a background medium properties factor to then obtain the model of the physical property or an update to an assumed model.
 3. The method of claim 1, wherein the seismic resolution volume is determined by ray tracing using velocity of a background medium model and using an assumed function of frequency for a seismic wavelet.
 4. The method of claim 2, wherein the model is determined from inversion of seismic data, said method further comprising: assuming an initial model of the subsurface region specifying a model parameter at discrete cell locations in the subsurface region; forming a mathematical objective function to measure misfit between measured seismic data and model-calculated seismic data; selecting a mathematical relationship that gives an adjustment, i.e. update, to the initial model that would reduce the misfit, said mathematical relationship relating said adjustment to a scaled gradient of the objective function, said gradient being with respect to said model parameter, the scaling comprising the four scale factors, i.e. the seismic resolution volume factor, the source illumination factor, the receiver illumination factor, and the background medium properties factor, all of which appear in the mathematical relationship as multiplicative factors that scale the gradient of the objective function to yield the adjustment to the model parameter; and using a computer to compute the adjustment from the mathematical relationship, and then updating the initial model with the computed adjustment.
 5. The method of claim 4, wherein the physical property, i.e. the model parameter, is either bulk modulus or density, or a combination of bulk modulus and density.
 6. The method of claim 4, wherein the mathematical relationship depends upon the physical property.
 7. The method of claim 4, wherein the background medium properties factor comprises bulk modulus raised to the fourth power divided by density squared when the physical property is bulk modulus, and comprises density squared when the physical property is density.
 8. The method of claim 4, wherein the receiver illumination factor when the physical property is bulk modulus is approximated by (1/8π)(ρ_(b)(r_(g))/ρ_(b)(r)), where ρ_(b)(r) is background density at location r and r_(g) is the receiver's location; and the receiver illumination factor when the physical property is density, I_(ρ,g)(r), is approximated by: ${I_{\rho,g}(r)} = {\frac{\rho_{b}\left( r_{g} \right)}{\rho_{b}(r)}{\frac{1}{8\pi\;{v_{p}^{2}(r)}}.}}$ where ν_(p)(r) is velocity at location r.
 9. The method of claim 4, further comprising repeating the method for at least one iteration, where in the initial model is replaced by the updated model from the previous iteration.
 10. The method of claim 9, wherein the objective function's functional form and the mathematical relationship's functional form are unchanged from one iteration to a next iteration.
 11. The method of claim 4, wherein the adjustment to the initial model is computed by minimizing the objective function using the objective function's Hessian resulting in an Hessian matrix, wherein off-diagonal elements of the Hessian matrix are ignored when outside of a seismic resolution volume.
 12. The method of claim 11, wherein off-diagonal elements of the Hessian matrix within a seismic resolution volume are assumed to be equal to a corresponding diagonal element, resulting in computing only diagonal elements.
 13. The method of claim 2, wherein the physical property is bulk modulus and the receiver illumination factor is approximated by an integral of a Green's function in free space over a surface defined by the survey's receiver spread.
 14. The method of claim 4, wherein the physical property is density and the receiver illumination factor is approximated by an integral of a gradient of a Green's function in free space over a surface defined by the survey's receiver spread.
 15. The method of claim 1, wherein the seismic resolution volume is approximated as a sphere based on an assumption of uniform wavenumber coverage.
 16. The method of claim 4, further comprising using the updated model to precondition the gradient in an iterative optimization technique.
 17. The method of claim 2, wherein the model is determined from reverse time migration of seismic images from the seismic data.
 18. The method of claim 17, wherein the physical property is bulk modulus.
 19. The method of claim 17, wherein the background medium properties factor comprises bulk modulus squared divided by density.
 20. The method of claim 17, wherein the receiver illumination factor is approximated by (1/8π)(ρ_(b)(r_(g))/ρ_(b)(r)), where ρ_(b)(r) is background density at location r and r_(g) is the receiver's location.
 21. The method of claim 17, wherein the migrated images seismic amplitudes are converted into difference bulk modulus or difference compressional wave velocity. 